Is W a subspace of the vector space?

[eyehategod]

W={(x1,x2,x3):x$$^{2}_{1}$$+x$$^{2}_{2}$$+x$$^{2}_{3}$$=0} , V=R^3

Is W a subspace of the vector space?
from what i understand for subspace to be a subspace it has to have two conditions:
1.must be closed under addition
2.must be closed under multiplication

so...
I pick a vector s=(s1,s2,s3) and a second vector t=(t1,t2,t3).

for the addition i get:
s+t=(s1+t1,s2+t2,s3+t3)//so its closed under addition

for multiplication i get:
cs=c(s1,s2,s3)=(cs1,cs2,cs3)//closed under multiplicartion

what i don't understand is how the condition x$$^{2}_{1}$$+x$$^{2}_{2}$$+x$$^{2}_{3}$$=0 would come into play. how do i use this condition in this problem?

 

[Dick]

You mean V=R^3, right? And your 'proof' just proves R^3 is a vector space. It doesn't say anything about W. What is the set W? Can you find a point in it? Can you describe a general point in W?

 

[radou]

Is W a subspace of the vector space?

Of which vector space? You should be more precise, although it is obvious you meant R^3.

from what i understand for subspace to be a subspace it has to have two conditions:
1.must be closed under addition
2.must be closed under multiplication

It is more practical to express this as one single condition; W is a subspace of V if and only if for any two vectors x, y from W, and any scalars a, b, ax + by is in W.

So, take any two vectors from W, let's say x = (x1, x2, x3) and y = (y1, y2, y3). Now write the linear combination ax + by, and see if the components of ax + by satisfy the condition for a vector to be in W. Also, while doing this, keep in mind that the components of x and y do satisfy this very condition!

 

[eyehategod]

my book says this:

If W is a nonempty subset of a vector space V, then W is a subspace of V if and only if the following closure conditions hold.
1.u and v are in W, then u+v is in w.
2if u is in W and c is a scalar, then cu is in W.

 

[Hurkyl]

So what does "x is in W" mean here?